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Simplifying 0 = -0.5 * 32x2 + 40x + 80 Multiply -0.5 * 32 0 = -16x2 + 40x + 80 Reorder the terms: 0 = 80 + 40x + -16x2 Solving 0 = 80 + 40x + -16x2 Solving for variable 'x'. Combine like terms: 0 + -80 = -80 -80 + -40x + 16x2 = 80 + 40x + -16x2 + -80 + -40x + 16x2 Reorder the terms: -80 + -40x + 16x2 = 80 + -80 + 40x + -40x + -16x2 + 16x2 Combine like terms: 80 + -80 = 0 -80 + -40x + 16x2 = 0 + 40x + -40x + -16x2 + 16x2 -80 + -40x + 16x2 = 40x + -40x + -16x2 + 16x2 Combine like terms: 40x + -40x = 0 -80 + -40x + 16x2 = 0 + -16x2 + 16x2 -80 + -40x + 16x2 = -16x2 + 16x2 Combine like terms: -16x2 + 16x2 = 0 -80 + -40x + 16x2 = 0 Factor out the Greatest Common Factor (GCF), '8'. 8(-10 + -5x + 2x2) = 0 Ignore the factor 8.Subproblem 1
Set the factor '(-10 + -5x + 2x2)' equal to zero and attempt to solve: Simplifying -10 + -5x + 2x2 = 0 Solving -10 + -5x + 2x2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -5 + -2.5x + x2 = 0 Move the constant term to the right: Add '5' to each side of the equation. -5 + -2.5x + 5 + x2 = 0 + 5 Reorder the terms: -5 + 5 + -2.5x + x2 = 0 + 5 Combine like terms: -5 + 5 = 0 0 + -2.5x + x2 = 0 + 5 -2.5x + x2 = 0 + 5 Combine like terms: 0 + 5 = 5 -2.5x + x2 = 5 The x term is -2.5x. Take half its coefficient (-1.25). Square it (1.5625) and add it to both sides. Add '1.5625' to each side of the equation. -2.5x + 1.5625 + x2 = 5 + 1.5625 Reorder the terms: 1.5625 + -2.5x + x2 = 5 + 1.5625 Combine like terms: 5 + 1.5625 = 6.5625 1.5625 + -2.5x + x2 = 6.5625 Factor a perfect square on the left side: (x + -1.25)(x + -1.25) = 6.5625 Calculate the square root of the right side: 2.561737691 Break this problem into two subproblems by setting (x + -1.25) equal to 2.561737691 and -2.561737691.Subproblem 1
x + -1.25 = 2.561737691 Simplifying x + -1.25 = 2.561737691 Reorder the terms: -1.25 + x = 2.561737691 Solving -1.25 + x = 2.561737691 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.25' to each side of the equation. -1.25 + 1.25 + x = 2.561737691 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + x = 2.561737691 + 1.25 x = 2.561737691 + 1.25 Combine like terms: 2.561737691 + 1.25 = 3.811737691 x = 3.811737691 Simplifying x = 3.811737691Subproblem 2
x + -1.25 = -2.561737691 Simplifying x + -1.25 = -2.561737691 Reorder the terms: -1.25 + x = -2.561737691 Solving -1.25 + x = -2.561737691 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.25' to each side of the equation. -1.25 + 1.25 + x = -2.561737691 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + x = -2.561737691 + 1.25 x = -2.561737691 + 1.25 Combine like terms: -2.561737691 + 1.25 = -1.311737691 x = -1.311737691 Simplifying x = -1.311737691Solution
The solution to the problem is based on the solutions from the subproblems. x = {3.811737691, -1.311737691}Solution
x = {3.811737691, -1.311737691}
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